Condense - Revision history

WikiSysop: /* Identities */

2018-04-15T21:14:03Z

Identities

← Older revision		Revision as of 17:14, 15 April 2018
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	==Identities==		==Identities==

	<apll>R ←→ < > R</apll>   for all <apll>R</apll> (see [[Dilate]] for the definition of monadic Right Caret)<br />		<apll> R ←→ < > R</apll>   for all Real (<apll>1==R</apll>) <apll>R</apll> (see [[Dilate]] for the definition of monadic Right Caret)<br />
	<apll>R ←→ <[X] >[X] R</apll>   for all ~~non-scalar~~ <apll>R</apll><br />		<apll> R ←→ <[X] >[X] R</apll>   for all Real (<apll>1==R</apll>) <apll>R</apll><br />
	<apll>R ←→ > < R</apll>   for all <apll>R</apll><br />		<apll> R ←→ > < R</apll>   for all <apll>R</apll> with <apll>(¯1↑⍴ R)∊1 2 4 8</apll><br />
	<apll>R ←→ >[X] <[X] R</apll>   for all non-scalar <apll>R</apll> with <apll>(⍴R)[X]∊1 2 4 8</apll><br />		<apll>1/R ←→ > < R</apll>   for all <apll>R</apll> with <apll>(¯1↑⍴1/R)∊1 2 4 8</apll><br />
			<apll> R ←→ >[X] <[X] R</apll>   for all non-scalar <apll>R</apll> with <apll>(⍴R)[X]∊1 2 4 8</apll><br />

	== Acknowledgements==		== Acknowledgements==

	<p>This symbol and its name were suggested by David A. Rabenhorst.</p>		<p>This symbol and its name were suggested by David A. Rabenhorst.</p>

Sudleyplace at 18:54, 15 April 2018

2018-04-15T18:54:00Z

← Older revision		Revision as of 14:54, 15 April 2018
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	</tr>		</tr>
	<tr>		<tr>
	<td><apll>X</apll> is an optional numeric singleton axis with <apll>X∊⍳⍴⍴R</apll>. If <apll>X</apll> is omitted, ~~it defaults~~ to the last axis.</td>		<td><apll>X</apll> is an optional numeric singleton axis with <apll>X∊⍳⍴⍴R</apll> &mddash; otherwise (including if <apll>R</apll> is a scalar), an <apll>AXIS ERROR</apll> is signalled. If <apll>[X]</apll> is omitted, the operation applies to the last axis.</td>
	</tr>		</tr>
	<tr>		<tr>
	<td><apll>R</apll> is an arbitrary Real numeric array ~~(BOOL~~, ~~INT~~, ~~FLT~~, ~~APA~~, ~~RAT~~, ~~VFP~~ — otherwise, <apll>DOMAIN ERROR</apll>~~) whose~~ <apll>X</apll>-axis length <apll>(⍴R)[X]</apll> is <apll>1</apll>, <apll>2</apll>, <apll>4</apll>, or <apll>8</apll> — otherwise, <apll>LENGTH ERROR</apll>.</td>		<td><apll>R</apll> is an arbitrary Real numeric array, i.e., Fixed-Precision Boolean, Integer, Floating Point, Arithmetic Progression Array, or Multiple-Precision Integer/Rational or Floating Point — otherwise, a <apll>DOMAIN ERROR</apll> is signalled. The <apll>X</apll>-axis length <apll>(⍴R)[X]</apll> must be <apll>1</apll>, <apll>2</apll>, <apll>4</apll>, or <apll>8</apll> — otherwise, a <apll>LENGTH ERROR</apll> is signalled.</td>
	</tr>		</tr>
	<tr>		<tr>
	<td><apll>Z</apll> is the corresponding Real or Hypercomplex array of shape <apll>(X≠⍳⍴⍴R)/⍴R</apll> using the ~~items~~ along the <apll>X</apll>-axis of <apll>R</apll> as the coefficients of the resulting Real or Hypercomplex array. If <apll>(⍴R)[X]</apll> is <apll>1</apll>, the result is ~~the~~ Real ~~array~~ <~~apll~~>~~((X≠⍳⍴⍴R)/⍴R)⍴R~~<~~/apll~~>~~, if~~ <apll>(⍴R)[X]</apll> is <apll>2</apll>, the result is a Complex ~~array~~, if <apll>(⍴R)[X]</apll> is <apll>4</apll>, the result is a Quaternion ~~array~~, ~~and if~~ <apll>(⍴R)[X]</apll> is <apll>8</apll>, the result is an Octonion ~~array~~.</td>		<td><apll>Z</apll> is the corresponding Real or Hypercomplex array of shape <apll>(X≠⍳⍴⍴R)/⍴R</apll> using the numbers along the <apll>X</apll>-axis of <apll>R</apll> as the coefficients of the resulting Real or Hypercomplex array. If
			<ul>
			<li><apll>(⍴R)[X]</apll> is <apll>1</apll>, the result is an array of Real numbers,</li>
			<li><apll>(⍴R)[X]</apll> is <apll>2</apll>, the result is an array of Complex numbers,</li>
			<li><apll>(⍴R)[X]</apll> is <apll>4</apll>, the result is an array of Quaternion numbers,</li>
			<li><apll>(⍴R)[X]</apll> is <apll>8</apll>, the result is an array of Octonion numbers.</li>
			</ul>
			</td>
	</tr>		</tr>
	</table>		</table>
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	<p>For example,</p>		<p>For example,</p>

	<apll>      <23~~<br />~~		<apll>      <23
	23~~<br />~~		23
	<10 20~~<br />~~		<10 20
	10<pn>J</pn>20~~<br />~~		10<pn>J</pn>20
	<2 4⍴⍳8~~<br />~~		<2 4⍴⍳8
	1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4 5<pn>i</pn>6<pn>j</pn>7<pn>k</pn>8<br />		1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4 5<pn>i</pn>6<pn>j</pn>7<pn>k</pn>8
	<2 8⍴(⍳8),⌽⍳8~~<br />~~		<[1] 2 4⍴⍳8
	1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<pn>l</pn>5<pn>ij</pn>6<pn>jk</pn>7<pn>kl</pn>8 8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5<pn>l</pn>4<pn>ij</pn>3<pn>jk</pn>2<pn>kl</pn>1~~<br />~~		1<pn>J</pn>5 2<pn>J</pn>6 3<pn>J</pn>7 4<pn>J</pn>8
	<[1] 2 8⍴(⍳8),⌽⍳8~~<br />~~		<2 8⍴(⍳8),⌽⍳8
	1<pn>J</pn>8 2<pn>J</pn>7 3<pn>J</pn>6 4<pn>J</pn>5 5<pn>J</pn>4 6<pn>J</pn>3 7<pn>J</pn>2 8<pn>J</pn>1~~<br />~~		1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<pn>l</pn>5<pn>ij</pn>6<pn>jk</pn>7<pn>kl</pn>8 8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5<pn>l</pn>4<pn>ij</pn>3<pn>jk</pn>2<pn>kl</pn>1
	⍴⎕←<2 3 1⍴⍳6~~<br />~~		<[1] 2 8⍴(⍳8),⌽⍳8
	1 2 3~~<br />~~		1<pn>J</pn>8 2<pn>J</pn>7 3<pn>J</pn>6 4<pn>J</pn>5 5<pn>J</pn>4 6<pn>J</pn>3 7<pn>J</pn>2 8<pn>J</pn>1
	4 5 6~~<br />~~		⍴⎕←<2 3 1⍴⍳6
	2 3~~<br />~~		1 2 3
	⍴⎕←<2 3 2⍴(⍳6),⌽⍳6~~<br />~~		4 5 6
	1<pn>J</pn>2 3<pn>J</pn>4 5<pn>J</pn>6~~<br />~~		2 3
	6<pn>J</pn>5 4<pn>J</pn>3 2<pn>J</pn>1<br />		⍴⎕←<2 3 2⍴(⍳6),⌽⍳6
	2 3<br />		1<pn>J</pn>2 3<pn>J</pn>4 5<pn>J</pn>6
	⍴⎕←<2 3 4⍴(⍳12),⌽⍳12~~<br />~~		6<pn>J</pn>5 4<pn>J</pn>3 2<pn>J</pn>1
	1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4~~ ~~   5<pn>i</pn>6<pn>j</pn>7<pn>k</pn>8 9<pn>i</pn>10<pn>j</pn>11<pn>k</pn>12~~<br />~~		2 3
	12<pn>i</pn>11<pn>j</pn>10<pn>k</pn>9 8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5 4<pn>i</pn>3<pn>j</pn>2<pn>k</pn>1~~<br />~~		⍴⎕←<[1] 2 3 2⍴(⍳6),⌽⍳6
	2 3~~<br />~~		1<pn>J</pn>6 2<pn>J</pn>5
	⍴⎕←<2 3 8⍴(⍳24),⌽⍳24~~<br />~~		3<pn>J</pn>4 4<pn>J</pn>3
	1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<pn>l</pn>5<pn>ij</pn>6<pn>jk</pn>7<pn>kl</pn>8         9<pn>i</pn>10<pn>j</pn>11<pn>k</pn>12<pn>l</pn>13<pn>ij</pn>14<pn>jk</pn>15<pn>kl</pn>16 17<pn>i</pn>18<pn>j</pn>19<pn>k</pn>20<pn>l</pn>21<pn>ij</pn>22<pn>jk</pn>23<pn>kl</pn>24~~<br />~~		5<pn>J</pn>2 6<pn>J</pn>1
	24<pn>i</pn>23<pn>j</pn>22<pn>k</pn>21<pn>l</pn>20<pn>ij</pn>19<pn>jk</pn>18<pn>kl</pn>17 16<pn>i</pn>15<pn>j</pn>14<pn>k</pn>13<pn>l</pn>12<pn>ij</pn>11<pn>jk</pn>10<pn>kl</pn>9   8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5<pn>l</pn>4<pn>ij</pn>3<pn>jk</pn>2<pn>kl</pn>1~~<br />~~		3 2
	2 3~~<br />~~		⍴⎕←<2 3 4⍴(⍳12),⌽⍳12
			1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4   5<pn>i</pn>6<pn>j</pn>7<pn>k</pn>8 9<pn>i</pn>10<pn>j</pn>11<pn>k</pn>12
			12<pn>i</pn>11<pn>j</pn>10<pn>k</pn>9 8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5 4<pn>i</pn>3<pn>j</pn>2<pn>k</pn>1
			2 3
			⍴⎕←<2 3 8⍴(⍳24),⌽⍳24
			1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<pn>l</pn>5<pn>ij</pn>6<pn>jk</pn>7<pn>kl</pn>8         9<pn>i</pn>10<pn>j</pn>11<pn>k</pn>12<pn>l</pn>13<pn>ij</pn>14<pn>jk</pn>15<pn>kl</pn>16 17<pn>i</pn>18<pn>j</pn>19<pn>k</pn>20<pn>l</pn>21<pn>ij</pn>22<pn>jk</pn>23<pn>kl</pn>24
			24<pn>i</pn>23<pn>j</pn>22<pn>k</pn>21<pn>l</pn>20<pn>ij</pn>19<pn>jk</pn>18<pn>kl</pn>17 16<pn>i</pn>15<pn>j</pn>14<pn>k</pn>13<pn>l</pn>12<pn>ij</pn>11<pn>jk</pn>10<pn>kl</pn>9   8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5<pn>l</pn>4<pn>ij</pn>3<pn>jk</pn>2<pn>kl</pn>1
			2 3
	</apll>		</apll>

	==Identities==		==Identities==

	<apll>R ←→ <~~[X]~~ >~~[X]~~ R</apll>   for all <apll>R</apll> (see [[Dilate]] for the definition of monadic Right Caret)<br />		<apll>R ←→ < > R</apll>   for all <apll>R</apll> (see [[Dilate]] for the definition of monadic Right Caret)<br />
	<apll>R ←→ >[X] <[X] R</apll>   for all <apll>R</apll> with <apll>(⍴R)[X]∊1 2 4 8</apll><br />		<apll>R ←→ <[X] >[X] R</apll>   for all non-scalar <apll>R</apll><br />
			<apll>R ←→ > < R</apll>   for all <apll>R</apll><br />
			<apll>R ←→ >[X] <[X] R</apll>   for all non-scalar <apll>R</apll> with <apll>(⍴R)[X]∊1 2 4 8</apll><br />

	== Acknowledgements==		== Acknowledgements==

	<p>This symbol and its name were suggested by David A. Rabenhorst.</p>		<p>This symbol and its name were suggested by David A. Rabenhorst.</p>

WikiSysop at 00:14, 12 April 2018

2018-04-12T00:14:42Z

← Older revision		Revision as of 20:14, 11 April 2018
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	<table border="0" cellpadding="5" cellspacing="0" summary="">		<table border="0" cellpadding="5" cellspacing="0" summary="">
	<tr>		<tr>
	<td valign="top"><apll>Z←<R</apll></td>		<td valign="top"><apll>Z←<R</apll>    or    <apll>Z←<[X] R</apll></td>
	<td></td>		<td></td>
	<td></td>		<td></td>
	<td>converts <apll>R</apll> to a Hypercomplex array if <apll>(≢R)∊2 4 8</apll> or a Real array if <apll>1~~=≢R~~</apll>.</td>		<td>converts <apll>R</apll> to a Hypercomplex array if <apll>(⍴R)[X]∊2 4 8</apll> or a Real array if <apll>(⍴R)[X]=1</apll>.</td>
	</tr>		</tr>
	</table>		</table>
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	</tr>		</tr>
	<tr>		<tr>
	<td><apll>R</apll> is an ~~arbitrary Real~~ numeric ~~array (BOOL, INT, FLT, APA, RAT, VFP — otherwise,~~ <apll>~~DOMAIN ERROR~~</apll>~~) whose number of columns (~~<apll>≢R</apll>) is ~~<apll>1</apll>, <apll>2</apll>, <apll>4</apll>, or <apll>8</apll> — otherwise~~, ~~<apll>LENGTH ERROR</apll>~~.</td>		<td><apll>X</apll> is an optional numeric singleton axis with <apll>X∊⍳⍴⍴R</apll>. If <apll>X</apll> is omitted, it defaults to the last axis.</td>
	</tr>		</tr>
	<tr>		<tr>
	<td><apll>Z</apll> is the corresponding Real or Hypercomplex array of shape <apll>~~¯1↓⍴R~~</apll> using the ~~columns~~ of <apll>R</apll> as the coefficients of the resulting Real or Hypercomplex array. If <apll>≢R</apll> is <apll>1</apll>, the result is the Real array <apll>(~~¯1↓⍴R~~)⍴R</apll>, if <apll>≢R</apll> is <apll>2</apll>, the result is a Complex array, if <apll>≢R</apll> is <apll>4</apll>, the result is a Quaternion array, and if <apll>≢R</apll> is <apll>8</apll>, the result is an Octonion array.</td>		<td><apll>R</apll> is an arbitrary Real numeric array (BOOL, INT, FLT, APA, RAT, VFP — otherwise, <apll>DOMAIN ERROR</apll>) whose <apll>X</apll>-axis length <apll>(⍴R)[X]</apll> is <apll>1</apll>, <apll>2</apll>, <apll>4</apll>, or <apll>8</apll> — otherwise, <apll>LENGTH ERROR</apll>.</td>
			</tr>
			<tr>
			<td><apll>Z</apll> is the corresponding Real or Hypercomplex array of shape <apll>(X≠⍳⍴⍴R)/⍴R</apll> using the items along the <apll>X</apll>-axis of <apll>R</apll> as the coefficients of the resulting Real or Hypercomplex array. If <apll>(⍴R)[X]</apll> is <apll>1</apll>, the result is the Real array <apll>((X≠⍳⍴⍴R)/⍴R)⍴R</apll>, if <apll>(⍴R)[X]</apll> is <apll>2</apll>, the result is a Complex array, if <apll>(⍴R)[X]</apll> is <apll>4</apll>, the result is a Quaternion array, and if <apll>(⍴R)[X]</apll> is <apll>8</apll>, the result is an Octonion array.</td>
	</tr>		</tr>
	</table>		</table>
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	<2 8⍴(⍳8),⌽⍳8<br />		<2 8⍴(⍳8),⌽⍳8<br />
	1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<pn>l</pn>5<pn>ij</pn>6<pn>jk</pn>7<pn>kl</pn>8 8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5<pn>l</pn>4<pn>ij</pn>3<pn>jk</pn>2<pn>kl</pn>1<br />		1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<pn>l</pn>5<pn>ij</pn>6<pn>jk</pn>7<pn>kl</pn>8 8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5<pn>l</pn>4<pn>ij</pn>3<pn>jk</pn>2<pn>kl</pn>1<br />
			<[1] 2 8⍴(⍳8),⌽⍳8<br />
			1<pn>J</pn>8 2<pn>J</pn>7 3<pn>J</pn>6 4<pn>J</pn>5 5<pn>J</pn>4 6<pn>J</pn>3 7<pn>J</pn>2 8<pn>J</pn>1<br />
	⍴⎕←<2 3 1⍴⍳6<br />		⍴⎕←<2 3 1⍴⍳6<br />
	1 2 3<br />		1 2 3<br />
Line 50:		Line 55:
	==Identities==		==Identities==

	<apll>R ←→ <>R</apll>   for all <apll>R</apll> (see [[Dilate]] for the definition of monadic Right Caret)<br />		<apll>R ←→ <[X] >[X] R</apll>   for all <apll>R</apll> (see [[Dilate]] for the definition of monadic Right Caret)<br />
	<apll>R ←→ ><R</apll>   for all <apll>R</apll> with <apll>(~~¯1↑⍴R~~)∊1 2 4 8</apll><br />		<apll>R ←→ >[X] <[X] R</apll>   for all <apll>R</apll> with <apll>(⍴R)[X]∊1 2 4 8</apll><br />

	== Acknowledgements==		== Acknowledgements==

	<p>This symbol and its name were suggested by David A. Rabenhorst.</p>		<p>This symbol and its name were suggested by David A. Rabenhorst.</p>

WikiSysop at 22:50, 9 April 2017

2017-04-09T22:50:12Z

← Older revision		Revision as of 18:50, 9 April 2017
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	23<br />		23<br />
	<10 20<br />		<10 20<br />
	~~10J20~~<br />		10<pn>J</pn>20<br />
	<2 4⍴⍳8<br />		<2 4⍴⍳8<br />
	~~1i2j3k4 5i6j7k8~~<br />		1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4 5<pn>i</pn>6<pn>j</pn>7<pn>k</pn>8<br />
	<2 8⍴(⍳8),⌽⍳8<br />		<2 8⍴(⍳8),⌽⍳8<br />
	~~1i2j3k4l5ij6jk7kl8 8i7j6k5l4ij3jk2kl1~~<br />		1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<pn>l</pn>5<pn>ij</pn>6<pn>jk</pn>7<pn>kl</pn>8 8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5<pn>l</pn>4<pn>ij</pn>3<pn>jk</pn>2<pn>kl</pn>1<br />
	⍴⎕←<2 3 1⍴⍳6<br />		⍴⎕←<2 3 1⍴⍳6<br />
	1 2 3<br />		1 2 3<br />
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	2 3<br />		2 3<br />
	⍴⎕←<2 3 2⍴(⍳6),⌽⍳6<br />		⍴⎕←<2 3 2⍴(⍳6),⌽⍳6<br />
	~~1J2~~ ~~3J4~~ ~~5J6~~<br />		1<pn>J</pn>2 3<pn>J</pn>4 5<pn>J</pn>6<br />
	~~6J5~~ ~~4J3~~ ~~2J1~~<br />		6<pn>J</pn>5 4<pn>J</pn>3 2<pn>J</pn>1<br />
	2 3<br />		2 3<br />
	⍴⎕←<2 3 4⍴(⍳12),⌽⍳12<br />		⍴⎕←<2 3 4⍴(⍳12),⌽⍳12<br />
	~~1i2j3k4~~    ~~5i6j7k8~~ ~~9i10j11k12~~<br />		1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4    5<pn>i</pn>6<pn>j</pn>7<pn>k</pn>8 9<pn>i</pn>10<pn>j</pn>11<pn>k</pn>12<br />
	~~12i11j10k9~~ ~~8i7j6k5~~ ~~4i3j2k1~~<br />		12<pn>i</pn>11<pn>j</pn>10<pn>k</pn>9 8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5 4<pn>i</pn>3<pn>j</pn>2<pn>k</pn>1<br />
	2 3<br />		2 3<br />
	⍴⎕←<2 3 8⍴(⍳24),⌽⍳24<br />		⍴⎕←<2 3 8⍴(⍳24),⌽⍳24<br />
	~~1i2j3k4l5ij6jk7kl8~~         ~~9i10j11k12l13ij14jk15kl16~~ ~~17i18j19k20l21ij22jk23kl24~~<br />		1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<pn>l</pn>5<pn>ij</pn>6<pn>jk</pn>7<pn>kl</pn>8         9<pn>i</pn>10<pn>j</pn>11<pn>k</pn>12<pn>l</pn>13<pn>ij</pn>14<pn>jk</pn>15<pn>kl</pn>16 17<pn>i</pn>18<pn>j</pn>19<pn>k</pn>20<pn>l</pn>21<pn>ij</pn>22<pn>jk</pn>23<pn>kl</pn>24<br />
	~~24i23j22k21l20ij19jk18kl17~~ ~~16i15j14k13l12ij11jk10kl9~~   ~~8i7j6k5l4ij3jk2kl1~~<br />		24<pn>i</pn>23<pn>j</pn>22<pn>k</pn>21<pn>l</pn>20<pn>ij</pn>19<pn>jk</pn>18<pn>kl</pn>17 16<pn>i</pn>15<pn>j</pn>14<pn>k</pn>13<pn>l</pn>12<pn>ij</pn>11<pn>jk</pn>10<pn>kl</pn>9   8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5<pn>l</pn>4<pn>ij</pn>3<pn>jk</pn>2<pn>kl</pn>1<br />
	2 3<br />		2 3<br />
	</apll>		</apll>

WikiSysop at 20:57, 9 April 2017

2017-04-09T20:57:44Z

← Older revision		Revision as of 16:57, 9 April 2017
Line 47:		Line 47:
	2 3<br />		2 3<br />
	</apll>		</apll>

			==Identities==

			<apll>R ←→ <>R</apll>   for all <apll>R</apll> (see [[Dilate]] for the definition of monadic Right Caret)<br />
			<apll>R ←→ ><R</apll>   for all <apll>R</apll> with <apll>(¯1↑⍴R)∊1 2 4 8</apll><br />

			== Acknowledgements==

	<p>This symbol and its name were suggested by David A. Rabenhorst.</p>		<p>This symbol and its name were suggested by David A. Rabenhorst.</p>

WikiSysop: Created page with "

2017-04-09T19:43:47Z

Created page with "<table border="1" cellpadding="5" cellspacing="0" rules="none" summary=""> <tr> <td> <table border="0" cellpadding="5" cellspacing="0" summary=""> <tr> <td val..."

New page
<table border="1" cellpadding="5" cellspacing="0" rules="none" summary="">
<tr>
<td>
<table border="0" cellpadding="5" cellspacing="0" summary="">
<tr>
<td valign="top"><apll>Z←<R</apll></td>
<td></td>
<td></td>
<td>converts <apll>R</apll> to a Hypercomplex array if <apll>(≢R)∊2 4 8</apll> or a Real array if <apll>1=≢R</apll>.</td>
</tr>
</table>
</td>
</tr>
<tr>
<td><apll>R</apll> is an arbitrary Real numeric array (BOOL, INT, FLT, APA, RAT, VFP — otherwise, <apll>DOMAIN ERROR</apll>) whose number of columns (<apll>≢R</apll>) is <apll>1</apll>, <apll>2</apll>, <apll>4</apll>, or <apll>8</apll> — otherwise, <apll>LENGTH ERROR</apll>.</td>
</tr>
<tr>
<td><apll>Z</apll> is the corresponding Real or Hypercomplex array of shape <apll>¯1↓⍴R</apll> using the columns of <apll>R</apll> as the coefficients of the resulting Real or Hypercomplex array. If <apll>≢R</apll> is <apll>1</apll>, the result is the Real array <apll>(¯1↓⍴R)⍴R</apll>, if <apll>≢R</apll> is <apll>2</apll>, the result is a Complex array, if <apll>≢R</apll> is <apll>4</apll>, the result is a Quaternion array, and if <apll>≢R</apll> is <apll>8</apll>, the result is an Octonion array.</td>
</tr>
</table>
<br />
<p>For example,</p>

<apll>      <23<br />
23<br />
      <10 20<br />
10J20<br />
      <2 4⍴⍳8<br />
1i2j3k4 5i6j7k8<br />
      <2 8⍴(⍳8),⌽⍳8<br />
1i2j3k4l5ij6jk7kl8 8i7j6k5l4ij3jk2kl1<br />
      ⍴⎕←<2 3 1⍴⍳6<br />
1 2 3<br />
4 5 6<br />
2 3<br />
      ⍴⎕←<2 3 2⍴(⍳6),⌽⍳6<br />
1J2 3J4 5J6<br />
6J5 4J3 2J1<br />
2 3<br />
      ⍴⎕←<2 3 4⍴(⍳12),⌽⍳12<br />
1i2j3k4    5i6j7k8 9i10j11k12<br />
12i11j10k9 8i7j6k5 4i3j2k1<br />
2 3<br />
      ⍴⎕←<2 3 8⍴(⍳24),⌽⍳24<br />
 1i2j3k4l5ij6jk7kl8         9i10j11k12l13ij14jk15kl16 17i18j19k20l21ij22jk23kl24<br />
24i23j22k21l20ij19jk18kl17 16i15j14k13l12ij11jk10kl9   8i7j6k5l4ij3jk2kl1<br />
2 3<br />
</apll>

<p>This symbol and its name were suggested by David A. Rabenhorst.</p>